Simultaneous envelopes for multivariate linear regression

R. Dennis Cook, Xin Zhang

Research output: Contribution to journalArticlepeer-review

39 Scopus citations


We introduce envelopes for simultaneously reducing the predictors and the responses in multivariate linear regression, so the regression then depends only on estimated linear combinations of X and Y. We use a likelihood-based objective function for estimating envelopes and then propose algorithms for estimation of a simultaneous envelope as well as for basic Grassmann manifold optimization. The asymptotic properties of the resulting estimator are studied under normality and extended to general distributions. We also investigate likelihood ratio tests and information criteria for determining the simultaneous envelope dimensions. Simulation studies and real data examples show substantial gain over the classical methods, like partial least squares, canonical correlation analysis, and reduced-rank regression. This article has supplementary material available online.

Original languageEnglish (US)
Pages (from-to)11-25
Number of pages15
Issue number1
StatePublished - Jan 2 2015

Bibliographical note

Publisher Copyright:
© 2015 American Statistical Association and the American Society for Quality.


  • Canonical correlations
  • Envelope model
  • Grassmann manifold
  • Partial least squares
  • Principal component analysis
  • Reduced-rank regression
  • Sufficient dimension reduction


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